We present a novel data smoothing and analysis framework for cortical thickness data defined on the brain cortical manifold. Gaussian kernel smoothing, which weights neighboring observations according to their 3D Euclidean distance, has been widely used in 3D brain images to increase the signal-to-noise ratio. When the observations lie on a convoluted brain surface, however, it is more natural to assign the weights based on the geodesic distance along the surface. We therefore develop a framework for geodesic distance-based kernel smoothing and statistical analysis on the cortical manifolds. As an illustration, we apply our methods in detecting the regions of abnormal cortical thickness in 16 high functioning autistic children via random field based multiple comparison correction that utilizes the new smoothing technique.